BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Charles Fougeron (Université de Paris) DTSTART:20210525T123000Z DTEND:20210525T133000Z DTSTAMP:20260423T021449Z UID:OWNS/46 DESCRIPTION:Title: Dy namics of simplicial systems and multidimensional continued fraction algor ithms\nby Charles Fougeron (Université de Paris) as part of One World Numeration seminar\n\n\nAbstract\nMotivated by the richness of the Gauss algorithm which allows to efficiently compute the best approximations of a real number by rationals\, many mathematicians have suggested generalisat ions to study Diophantine approximations of vectors in higher dimensions. Examples include Poincaré's algorithm introduced at the end of the 19th c entury or those of Brun and Selmer in the middle of the 20th century. Sinc e the beginning of the 90's to the present day\, there has been many works studying the convergence and dynamics of these multidimensional continued fraction algorithms. In particular\, Schweiger and Broise have shown that the approximation sequence built using Selmer and Brun algorithms converg e to the right vector with an extra ergodic property. On the other hand\, Nogueira demonstrated that the algorithm proposed by Poincaré almost neve r converges. \n\nStarting from the classical case of Farey's algorithm\, w hich is an "additive" version of Gauss's algorithm\, I will present a comb inatorial point of view on these algorithms which allows to us to use a ra ndom walk approach. In this model\, taking a random vector for the Lebesgu e measure will correspond to following a random walk with memory in a labe lled graph called symplicial system. The laws of probability for this rand om walk are elementary and we can thus develop probabilistic techniques to study their generic dynamical behaviour. This will lead us to describe a purely graph theoretic criterion to check the convergence of a continued f raction algorithm.\n LOCATION:https://researchseminars.org/talk/OWNS/46/ END:VEVENT END:VCALENDAR